The Algebra of Assortative Encounters

and the Evolution of Cooperation

Theodore C. Bergstrom1

University of California Santa Barbara

February 27, 2001

1This paper has benefited from comments by participants in the EvolutionaryGame Theory workshop held at the University of Odense, Denmark in September2000 and a workshop on Groups, Multi-level Selection and Economic Dynamics,held at the Santa Fe Institute in January of 2001. Special thanks for helpful remarksare due to Marc Feldman, Hillard Kaplan,Thorbjørn Knudsen, Chris Proulx, andDavid Sloan Wilson.

1 Introduction

Consider a population of individuals who meet and play Prisoners’ Dilemma.Players do not deliberately choose their strategies, but are “programmed” toplay either cooperate or defect. In Prisoners’ Dilemma, everyone gets a higherpayoff from playing defect than from playing cooperate, but everyone getsa higher payoff from playing against a cooperator than against a defector.If both types are equally likely to play against cooperators, then defectorswill get higher average payoffs than cooperators. But if cooperators aremore likely to encounter cooperative opponents, then the lower payoff fromplaying cooperate may be compensated by a higher probability of playingagainst a cooperating opponent.

This paper explores the quantitative relation between non-random, as-sortative matching and the maintenance of cooperative behavior under evo-lutionary dynamics. Section 2 develops an “algebra of assortative encoun-ters” and defines an index of assortativity of encounters. This section thenapplies the index of assortativity to study population dynamics under theassumption that the type with the higher payoff increases its proportion inthe population. Section 3 calculates the index of assortativity for games be-tween relatives where it is assumed that behavior of offspring is determinedby either cultural or genetical inheritance. This section shows the logicalconnection between the index of assortativity and Hamilton’s theory of kinselection [5]. Section 4 explores the index of assortativity and populationdynamics for a model in which players select their partners, using partiallyinformative cues about each others’ types.

2 Assortative Encounters in Prisoners’ Dilemma

2.1 The Payoff Matrix

We assume that there is a large population of players who meet other playersaccording to some specified encounter rule. When two players meet, theyplay a game of Prisoners’ Dilemma.1 Each individual is programmed forone of two possible strategies, cooperate (C) or defect (D). The payoffs forthe game are given in Table 1. This game is a Prisoners’ Dilemma when thepayoffs satisfy the inequalities, T > R > P > S.

1The algebra found here applies to any symmetric two-person, two-strategy game, in-cluding chicken, battle of the sexes, or the stag-hunt game. For concreteness, our discussionwill focus on Prisoners’ Dilemma games.

1

Table 1: A Prisoners’ Dilemma Game

Player 1

Player 2C D

C R, R S, T

D T, S P, P

2.2 The Algebra of Encounters

The index of Assortativity

Let x denote the proportion of the population that are cooperators (C-strategists) and 1 − x the proportion of defectors (D-strategists). Eachmember of the population randomly encounters one other member of thepopulation. The probability that an individual encounters a cooperator de-pends, in general, both on that individual’s own type and on the proportionof cooperators in the population. Let p(x) be the conditional probabilitythat one encounters a cooperator, given that one is a cooperator and letq(x) be the conditional probability that one encounters a cooperator, giventhat one is a defector.

The fraction of all encounters between two individuals in which a co-operator meets a defector is x (1− p(x)). The fraction of all encounters inwhich a defector meets a cooperator is (1 − x)q(x). Since these are justtwo different ways of counting the same encounters, we have the following“parity equation”

x (1− p(x)) = (1− x)q(x) (1)

Let us define the index of asssortativity a(x) to be the difference betweenthe probability that a C-strategist meets a C-strategist and the probabilitythat a D-strategist meets a C-strategist. Thus we have

a(x) = p(x)− q(x) (2)

2

Since (1− q(x)) − (1− p(x)) = p(x) − q(x) = a(x), it follows that a(x)is also the difference between the probability that a D-strategist meets aD-strategist and the probability that a C-strategist meets a D-strategist.Thus for either type, a(x) is the difference between the probability that onemeets one’s own type and the probability that a member of the other typemeets one’s own type.

Rearranging terms in Equation 1, and substituting from the definitionin 2, we find that

q(x) = x[1− (p(x)− q(x))] (3)= x (1− a(x)) . (4)

From Equations 2 and 4, it follows that

p(x) = a(x) + x (1− a(x)) . (5)

A Historical Digression on Measuring Assortativity

Sewall Wright [13] defined the assortativeness of mating with respect to agiven trait as the coefficient of correlation m between the two mates withrespect to their possession of the trait. Cavalli-Sforza and Feldman [3] inter-pret this correlation as follows. “The population is conceived of as containinga fraction (1 −m) that mates at random and a complementary fraction mwhich mates assortatively.” With this interpretation, if the population fre-quency of a type is x, then the probability that an individual of that typemates an individual of its own type is p(x) = m + x(1−m). The parameterm can be shown to be the coefficient of correlation between indicator ran-dom variables for possession of the trait for pairs of mates.2 From Equation5, we see that in the special case where a(x) = m is a constant, Wright’scoefficient of correlation and Cavalli-Sforza’s and Feldman’s “fraction of thepopulation that mates assortatively” are formally equivalent to the index ofassortativity defined in this paper.

2Let Ii be a random variable that takes on value 1 if individual i is of the given typeand 0 otherwise. For two partners, 1 and 2, the correlation coefficient between I1 and I2 isdefined to be (E(I1I2)−E(I1)E(I2))/(σ1σ2) where σi is the standard deviation of Ii. Now

E(I1I2) = xp(x), and for i = 1, 2, E(Ii) = x and σi =√

x(1− x). Making the appropriate

substitutions, we find that the correlation coefficient is (xp(x)− x2)/x(1− x) = m.

3

2.3 Evolutionary Dynamics and Comparing Payoffs

We assume that at any time, the growth rate of the proportion of cooperatorsin the population is positive, zero, or negative, depending on whether theexpected payoff of cooperators is higher than, equal to, or lower than theexpected payoff of defectors. Weibull [11] defines this to be the assumptionof payoff monotonicity.3 To study payoff monotone dynamics, we simplycompare the expected payoffs of cooperators and defectors.

With probability p(x) a cooperator meets another cooperator and getspayoff R and with probability 1 − p(x) a cooperator meets a defector andgets payoff S. Therefore the expected payoff to a cooperator is:

p(x)R + (1− p(x)) S = S + p(x)(R− S)= S + a(x)(R− S) + x (1− a(x)) (R− S) (6)

where the last expression is obtained by substituting for p(x) from Equation5. Similar reasoning shows that the expected payoff to a defector is

q(x)T + (1− q(x)) P = P + x (1− a(x)) (T − P ). (7)

Let us define δ(x) to be the difference between the expected payoff of acooperator and that of a defector when the proportion of cooperators in thepopulation is x. Substracting Equation 7 from Equation 6, we have

δ(x) = S − P + a(x)(R− S) + x(1− a(x))[(R + P )− (S + T )]. (8)

The function δ(·) plays the starring role in our study of payoff monotonicdynamics, since for all x between 0 and 1 the sign of the growth rate of theproportion of cooperators is the same as the sign of δ(x).

2.4 Prisoners’ Dilemma Games with Additive Payoffs

We can define a special class of Prisoners’ Dilemma games in which the ben-efits of being helped and the costs of helping the other player are “additive.”The restrictive assumption in the additive case is that the cost to one player

3A much-studied special case of payoff monotone dynamics is replicator dynamics inwhich the growth rate of the population share using a strategy is proportional to thedifference between the average payoff to that strategy and the average payoff in the entirepopulation. [11]. The results found in this paper do not require the special structure ofreplicator dynamics.

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of helping the other is independent of whether the help is reciprocated, andthe benefit that a player gets from being helped is independent of whether heis also helping. In this class of games, each player has the option of helpingthe other player (the C-strategy) or not helping (the D-strategy). A playerwho helps bears a cost of c and confers a benefit of b on the other player,where 0 < c < b. If each player helps the other, then both get a benefit of band both bear costs of c. Thus the payoff to mutual cooperation is R = b−c.If one player helps and the other does not, the player who helps bears a costof c and gets no benefit and the player who doesn’t help gets a benefit of band bears no cost. Thus S = −c and T = b. Finally if neither helps, both getpayoffs P = 0. These payoffs satisfy the inequalities necessary for the gameto be a Prisoners’ Dilemma, since S = −c < P = 0 < R = b− c < T = b.

For additive Prisoners’ Dilemma games, we have R + P = S + T = b− cand therefore

(R + P )− (S + T ) = 0 (9)

Therefore for additive Prisoners’ Dilemma games, Equation 8 for the dif-ference between the expected payoffs of cooperators and defectors simplifiesto

δ(x) = S − P + a(x)(R− S) = a(x)b− c. (10)

According to Equation 10, with additive payoffs, cooperators will dobetter or worse than defectors depending on whether the product of theindex of assortativity times the benefits conferred by help is greater than orless than the cost of helping.

2.5 Prisoners’ Dilemma Games with Non-Additive Payoffs

It is important to understand that not all Prisoners’ Dilemma games haveadditive payoffs and that the evolutionary dynamics can be qualitatively dif-ferent in the non-additive cases. We will show that every Prisoners’ Dilemmagame is equivalent to a game in which the two players each contribute costlyeffort to produce a joint output that is shared equally between them. Withthis parameterization, the class of additive Prisoners’ Dilemma games corre-sponds to those in which the joint “production function” exhibits constantreturns to scale.

Working and Shirking in a Game of Shared Output

Let us describe the family of PD games with shared output as follows. Eachplayer, i, can contribute either work, in which case si = 1 or shirk, in which

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case, si = 0. Total output is divided equally between the players, regardlessof their work effort. We will assume that the amount of output produced isgiven by a production function Φ that takes the following form:

Φ(s1, s2) = 2b(s1 + s2) + 2ks1s2 (11)

where k is a parameter such that −∞ < k < c. The cost of effort foreach i is assumed to be (c + b)si. With this production function, totaloutput when both players work will be greater than, equal to, or less thantwice total output when one works and one shirks, depending on whetherk > 0, k = 0, or k < 0. These three cases correspond respectively toproduction processes that are superadditive (increasing returns to scale),additive, (constant returns to scale), and subadditive (decreasing returns toscale).

The payoff to player 1 for a game of shared outputs is given by thefunction

Π(s1, s2) =12Φ(s1, s2)− (c + b)s1

= b(s1 + s2) + ks1s2 − (c + b)s1. (12)

For this game we see that T = Π(0, 1) = b, R = Π(1, 1) = 2b + k− (c + b) =b+k− c, P = Π(0, 0) = 0, and S = Π(1, 0) = b− (b+ c) = −c. For all k < c,these payoffs satisfy the inequalities S < P < R < T . A simple calculationshows that (R + P ) − (S + T ) = k. The game has additive payoffs onlywhere k = 0 and in this case, R + P = S + T .

To see that every Prisoner’s Dilemma game can be described by thisparameterization of a PD game with shared output, notice that withoutchanging the dynamics, we can normalize the game so that P = 0.4 Forany Prisoner’s Dilemma game with S < P = 0 < R < T , we can show thatthere is exactly one set of parameters for a PD game with shared outputfor which these are the payoffs. In particular, a Prisoner’s Dilemma gamefor which the parameters are S, P = 0, R, and T will be equivalent toa PD game with shared output if and only if the parameters b, k, and cof the production function and cost function satisfy b = T , c = −S, andk = R− (S + T ).

4Since we assume that the population dynamics of Prisoners’ Dilemma game dependson expected payoffs, these dynamics will be unchanged if we subtract a constant amountfrom each payoff so as to make P = 0.

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2.6 Dynamics with a Constant Index of Assortativity

As we will show, there are interesting applications in which the index ofassortativity, a(x) = a is constant with respect to x. When a(x) is constant,the dynamics are especially simple, since the expression for δ(x) Equation 8is seen to be linear in x.

The dynamics are further simplified if payoffs are additive and a(x) isconstant. Then, δ(x) = ab − c is constant for all x. When this is the case,equilibrium must be unique and stable. If ab > c, cooperators always havehigher expected payoffs than defectors and the only equilibrium is x = 1. Ifab < c, defectors always have higher payoffs and the only stable equilibriumis x = 0.

For non-additive Prisoners’ Dilemma games when a is constant, the qual-itative dynamics are determined by the signs of

δ(0) = [aR + (1− a)S]− P and (13)

δ(1) = R− [aP + (1− a)T ] (14)

There are four distinct possibilities.

• The case δ(0) > 0 and δ(1) > 0 occurs when aR + (1 − a)S > P andR > aP + (1 − a)T . In this case, δ(x) > 0 for all x between 0 and1. Therefore cooperators always have higher expected payoffs thandefectors and there is a unique stable equilibrium with x = 1.

• The case δ(0) < 0 and δ(1) < 0 occurs when aR + (1 − a)S < Pand R < aP + (1 − a)T . In this case, δ(x) < 0 for all x between 0and 1. Therefore defectors always have a higher expected payoff thancooperators and there is a unique stable equilibrium with x = 0.

• Figure 1 shows the graph of δ(x) when δ(0) > 0 and δ(1) < 0. If theseinequalities are satisfied, cooperators have a higher expected payoffthan defectors when cooperators are rare and defectors have a higherexpected payoff than cooperators when defectors are rare. We see fromFigure 1 that in this case, the unique equilibrium is a polymorphicpopulation that includes some cooperators and some defectors. Thiscase occurs when aR + (1− a)S < P and R < aP + (1− a)T .

• Figure 2 shows the graph of δ(x) when δ(0) < 0 and δ(1) > 0. If theseinequalities are satisfied, cooperators have a higher expected payoffthan defectors when x = 1 and defectors have a higher expected payoff

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Figure 1: Unique Polymorphic Equilibrium

δ(x)

←→x

0 1

than cooperators when x = 0. In this case, there are two monomorphicequilibria, one with cooperators only and one with defectors only. Thiscase occurs when aR + (1− a)S < P and R > aP + (1− a)T .

Figure 2: Two Stable Equilibria; x = 0 and x = 1

δ(x)

→←x

0 1

Interpretation for games of shared outputs

If we parameterize Prisoners’ Dilemma games as games of shared outputswith production function Φ(s1, s2) = 2b(s1 + s2) + 2ks1s2 and cost of effort(c + b)si, then we have

δ(0) = [aR + (1− a)S]− P = ab + ak − c and (15)

δ(1) = R− [aP + (1− a)T ] = ab + k − c (16)

In the additive case, where k = 0, δ(1) = δ(0) = ab − c. Where k 6= 0,we have δ(1) − δ(0) = k(1 − a). Therefore δ(1) − δ(0) is positive in whenthe production function is superadditive and negative where the productionfunction is subadditive. We see from Figures 1 and 2 that if a(x) is constant,a stable polymorphic equilibrium is possible only in the case of subadditiveproduction and two distinct stable monomorphic equilibria are possible onlyin the case of superadditive production.

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3 Kin-selection and Assortative Matching

Biologists are familiar with Equation 10 under the name Hamilton’s rule.According to W.G. Hamilton’s [5] theory of kin selection, natural selectionwill favor individuals who are willing to help a genetic relative if and only ifthe product of the benefits from helping times the coefficient of relatednessbetween the two relatives exceeds the cost of helping. Hamilton defines thecoefficient of relatedness between two individuals as the probability thatthe alleles found in a randomly selected genetic locus in the two individualsare inherited from the same ancestor. In a population without inbreeding,the coefficient of relatedness is one half for full siblings, one fourth for halfsiblings, and one eighth for first cousins.

Hamilton’s coefficient of relatedness plays the same formal role in histheory as our “index of assortativeness” for the case of additive Prisoners’Dilemma games. We will see that this is no accident. The examples belowshow how the index of assortativeness can be calculated for siblings undera variety of assumptions about mechanisms of cultural or genetic inheri-tance. It is interesting to notice that in all of these examples, the index ofassortativeness, a(x), turns out to be constant with respect to x.

3.1 Sexual Haploid Siblings

Let us consider a population in which children play a Prisoners’ Dilemmagame with their siblings. Each child has two parents and inherits its strategyfor this game from one of its parents, chosen at random. Since for mostanimals, including humans, genetic inheritance is sexual diploid rather thansexual haploid, the sexual haploid model is more suitable as a model ofculturally transmitted rather than genetically transmitted characteristics.5

A symmetric case with monogamy and random mating

Suppose that parents mate monogamously and randomly with respect tothe strategy that they play with siblings. Assume that each child is equallylikely to inherit its strategy from its mother or its father and that the parentcopied by one child is statistically independent of that copied by its siblings.

Where x is the proportion of cooperators in the entire population, let uscalculate the probability that a randomly chosen sibling of a cooperator is

5A sexual diploid individual carries two alleles, one inherited from its mother and oneinherited from its father, in each genetic locus. The two alleles found at a genetic locusthen determine the effect of this locus on the individual’s characteristics.

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also a cooperator. With probability 1/2, the sibling will have the same rolemodel as the cooperator, in which case the sibling will surely be a cooperator.With probability 1/2, the sibling will have a different role model. Since theparents are assumed to be mated randomly with respect to their strategiestoward siblings, the probability that the other parent is a cooperator willbe x. Therefore the probability that a cooperator has a sibling who is alsoa cooperator is

p(x) =12

+12x (17)

If a child is a defector, then its sibling will be a cooperator only if thesibling’s role model is different from the defector’s. With probability 1/2,the two siblings will have different role models, and given that they havedifferent role models, the probability that the other parent is a cooperatoris x. Therefore we have

q(x) =12x (18)

anda(x) = p(x)− q(x) =

12

(19)

Assortative mating and extra-familial influence

Suppose that all else is as in the previous section, but that there is assorta-tive mating between parents. The degree of assortativeness in mating canbe defined with the same formalism that we used to define the degree ofassortativeness in matching for game-playing encounters. In particular, thedegree of assortativeness in mating is a number m between 0 and 1, such thatwhen the proportion of C-strategists in the population is x, the probabilitythat a C-strategist mates with another C-strategist is x + m(1− x).

Now we can calculate the probablility p(x) that a random sibling of acooperator child is also a cooperator. A cooperator child has at least onecooperator parent, whose strategy the child imitates. With probability 1/2,the child’s sibling imitates the same parent and is also a cooperator. Withprobability 1/2, the sibling imitates the other parent. Given the degreem of assortativeness in mating, the other parent will be a cooperator withprobability x + m(1− x). Therefore the probability that a random sib of acooperator child is also a cooperator is

p(x) =12

+x + m(1− x)

2=

1 + m

2+

(1−m)x2

(20)

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A sibling of a defector child is a cooperator only if the two siblings havedifferent role models and if the sibling’s defector parent is mated with acooperator. The probability of this event is

q(x) =(1−m)x

2. (21)

Therefore the of assortativity between children is simply

a(x) = p(x)− q(x) =1 + m

2(22)

Another wrinkle that can be added to this calculation is to suppose thatwith some probability v, a child copies neither of its parents but ratherchooses a random member of the population to copy. In this case it is nothard to show that

p(x) = v

(12

+x + m(1− x)

2

)+ (1− v)x (23)

q(x) = v(1−m)x

2+ (1− v)x. (24)

and hencea(x) = p(x)− q(x) =

v(1 + m)2

(25)

Some asymmetry and some polygamy

Consider a partially polygamous population where the probability that twochildren of the same mother have the same father is µ. Suppose that childrencopy their strategy from their mother with probability λ and their fatherwith probability 1− λ.

We first calculate the probability that a random sibling of a cooperatoris a cooperator. This can happen in any of the following ways: both siblingsinherit their strategies from their mother, both siblings inherit their strate-gies from the same father, or the two siblings inherit their strategies fromdifferent parents. Adding the relevant probabilities, we find that

p(x) = λ2 + (1− λ)2µ + 2λ(1− λ)x + (1− λ)2(1− µ)x (26)

A defector’s sibling will be a cooperator only if the two siblings inherittheir strategies from different parents and the defector’s sibling inherits itsstrategy from a cooperator. The probability of this happening is

q(x) = 2λ(1− λ)x + (1− λ)2(1− µ)x (27)

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Thus we have

a(x) = p(x)− q(x) = λ2 + (1− λ)2µ (28)

3.2 Sexual Diploid Siblings

Most sexually-reproducing creatures, including humans, reproduce by diploidrather than haploid genetics. In each genetic locus, an individual carries twoalleles, one inherited from its father and one inherited from its mother. Theallele inherited from each parent is randomly selected from the two allelescarried by that parent. An individual’s genotype is a specification of the twogenes that it carries. If there are two allele types A and a, then there arethree possible genotypes, AA, Aa, and Aa. An individual who carries twoalleles of the same type is said to be a homozygote and one who carries twodifferent types of alleles is said to be a heterozygote. The allele A is saidto be dominant over allele a if heterozygotes of genotype Aa use the samestrategy as homozygotes of type AA.

The dynamics of polymorphic equilibria in diploid populations is com-plex and in general seems not to be amenable to the simple methods dis-cussed in this paper. However, our methods work well to characterize neces-sary conditions for a monomorphic equilibrium to be stable against dominantmutant alleles. A monomorphic population is one in which all individuals,except for rare mutants, are homozygotes of the same type. A monomor-phic equilibrium is stable against dominant mutant alleles if, so long as thenumber of mutant alleles is small, the expected payoff to players who carrythe mutant allele is smaller than that of players who are homozygotes withthe normal allele.

Suppose that there are two types of alleles C and D, and that individualsof genotype CC play cooperate, while individuals of genotype DD play de-fect in games with their siblings. Let x denote the fraction of C genes in thepopulation, let p(x) be the probability that a random sibling of a child whoplays cooperate has a sibling plays cooperate and let q(x) be the probabilitythat a random sibling of a child who plays defect will play cooperate. Wecan calculate an index of assortativity, a(x) for the limiting values of x = 1and x = 0. With these values in hand, we can determine conditions underwhich a monomorphic equilibrium population of cooperators or of defectorsis stable.

Consider a monomorphic population of CC genotypes. This populationis stable only if individuals who carry a dominant mutant D allele receive

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a lower expected payoff than the normal CC genotypes. Let us assumethat mating is sufficiently random with respect to genotype so that whenthe fraction of carriers of mutant genes is small, the probability that anindividual who carries the mutant allele mates with another carrier of themutant allele is nearly zero. Then so long as the mutant D allele is rare,almost every child that is born with the mutant allele has one heterozygoteCD parent and one homozygote CC parent.

When x is nearly one, almost every cooperator child has two homozygoteCC parents, in which case it’s sibling is sure to be a cooperator. Therefore

limx→1

p(x) = 1 (29)

When x is nearly one, almost every defector child has one homozygote CCparent and one heterozygote CD parent. The defector child’s sibling is sureto inherit a C allele from its CC parent, but the probability is only 1/2 thatit will receive a C allele from the CD parent. Therefore the probability thata defector child has a cooperator parent is 1/2. Thus

limx→1

q(x) =12

(30)

It follows thatlimx→1

a(x) = a(1) = 1− 12

=12

(31)

Equations 8, 30, and 29 together imply that the limiting value of thedifference in the expected payoffs of cooperators and of defectors will be

limx→1

δ(x) = a(1)R + (1− a(1))S − P + (1− a(1))[(R + P )− (S + T )]

= R− 12(P + T ). (32)

From Equation 32, it follows that a monomorphic population of C alleleswill be stable against invasion by dominant mutant D alleles if and only ifR > (P + T )/2.

Now let us explore conditions under which a monomorphic population ofD alleles is stable against invasion by dominant mutant C alleles. When x isnearly zero, a cooperator child will almost certainly have one heterozygoteCD parent and one homozygote DD parent. Its sibling will certainly inherita D allele from the homozygote parent, and with probability 1/2, the siblingwill inherit a C gene from the heterozygote parent. Thus the sibling will beof genotype CD or DD, each with probability 1/2. Where the C allele is

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dominant, this means that the sibling will cooperate with probability 1/2.Therefore we have

limx→0

p(x) =12. (33)

Where x is nearly zero, almost all defector offspring will have two homozy-gote DD parents, and thus their siblings will almost certainly be defectors.It follows that

limx→0

q(x) = 0 (34)

and thatlimx→0

a(x) = a(0) =12− 0 =

12

(35)

In a population consisting almost entirely of DD homozygotes, the dif-ference in the expected payoff of cooperators, who are almost always CDgenotypes, and of the defectors who are DD genotypes will be

limx→0

δ(x) = a(0)R + (1− a(0))S − P =12(R + S)− P. (36)

From Equation 36, it follows that a monomorphic population of D alleleswill be stable against invasion by dominant mutant C alleles if and only if(R + S)/2 < P .

We have shown that the condition for a monomorphic population ofcooperators to be an equilibrium is R > (P + T )/2 and the condition fora monomorphic population of defectors to be a stable equilibrium is (R +S)/2 < P . In the case of Prisoners’ Dilemma games with additive payoffs,we have R = b − c, P = 0, and T = b. In this case, the condition for amonomorphic population of cooperators to be an equilibrium is that b/2 >c and the condition for a monomorphic population of defectors to be anequilibrium is b/2 < c. Thus, in the additive case where b/2 6= c, wehave either a monomorphic equilibrium of cooperators or a monomorphicpopulation of defectors but not both. Which type of behavior prevails inequilibrium is determined by “Hamilton’s Rule.”

Equilibria for games of shared output

Recall that we defined a class of Prisoners’ Dilemma games that can berepresented as games of shared outputs with production function Φ(s1, s2) =2b(s1 + s2) + 2ks1s2 and cost of effort (c + b)si. Since each player receiveshalf of the total output and each pays the cost of his own output, the payofffunction for player 1 is Π(s1, s2) = b(s1 + s2) + ks1s2 − (c + b)s1. For this

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game, T = Π(0, 1) = b, R = Π(1, 1) = b + k − c, P = Π(0, 0) = 0 and S =Π(1, 0) = −c. Depending on the parameter values, b, c, and k, there maybe no stable monomorphic equilibrium, there may be a single monomorphicequilibrium consisting of either cooperators or of defectos, or there may betwo distinct stable monomorphic equilibria, one with cooperators only andone with defectors only. . The outcome x = 1 is a stable equilibrium ifR − (P + T )/2 > 0 and is not a stable equilibrium if this inequality isreversed. For games with shared output, this condition is equivalent to

b/2− c > −k (37)

The outcome x = 0 is a stable equilibrium if (R + S)/2 − P < 0 and nota stable equilibrium if this inequality is reversed. Equivalently, x = 0 is astable equilibrium if

b/2− c < −k/2. (38)

Ignoring the boundary cases (where the inequality in either or both ofthe expressions 37 or 38 is replaced by an equality) we have four possiblecases:

• If b/2 − c > max{−k,−k/2} then 37 is satisfied and the inequalityin 38 is reversed. In this case there exists a stable equilbrium withcooperators only and no stable equilibrium with defectors only.

• If b/2 − c < min{−k,−k/2} then 38 is satisfied and the inequality in37 is reversed, In this case exists a stable equilibrium with defectorsonly and no stable equilibrium with defectors only.

• if −k < b/2− c < −k/2, then both of the inequalities 37 and 38 holdand therefore there are two distinct stable monomorphic equilibria,one with cooperators only and one with defectors only. Since this caserequires that k/2 < k, it is possible only where k > 0, which meansthat production is superadditive.

• if −k/2 < b/2 − c < −k, then neither of the inequalities 37 and 38hold and there are no stable monomorphic equilibria. Since this caserequires that k < k/2, it is possible only where k < 0, which meansthat production is subadditive.

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4 Assortative Matching with Partner Choice

4.1 A model of labelling

Interesting possibilities for assortative matching arise when players havesome choice about their partners. In a game of Prisoners’ Dilemma, everyonewould prefer to be matched with a cooperator rather than with a defector.Let us assume that each player can make a fixed number of matches, thatsearch costs are negligible, and that a match requires mutual consent of thetwo matched players. If players’ types were observable with perfect accuracy,then the only equilibrium outcome would have cooperators matched onlywith cooperators and defectors with defectors. In this case, a(x) = 1 forall x and therefore cooperators receive a payoff R while defectors receiveP < R. Thus a population consisting only of cooperators would be the onlystable equilbrium.

But suppose that detection is less than perfectly accurate. Players arelabelled with an imperfect indicator of their type. (e.g. reputation based onpartial information, behavioral cues, or a psychological test ) Assume thatwith probability α > 1/2, a cooperator is correctly labelled as a cooperatorand with probability 1 − α is mislabelled as a defector with probability .Assume that with probability β > 1/2, a defector is correctly labelled andwith probability 1− β is mislabelled as a cooperator.

Everyone sees the same labels, so that at the time when players choosepartners there are only two distinguishable types: players who appear to becooperators and players who appear to be defectors. Although everyone re-alizes that the indicators are not entirely accurate, everyone prefers to matchwith an apparent cooperator rather than an apparent defector. Therefore,with voluntary matching, apparent cooperators will all be matched withapparent cooperators and apparent defectors with apparent defectors.

4.2 Calculating the index of assortativity

The proportion of actual cooperators among apparent cooperators is

Cc(x) =αx

αx + (1− β)(1− x)(39)

and the proportion of actual cooperators among apparent defectors is

Cd(x) =(1− α)x

(1− α)x + β(1− x)(40)

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There are two possible ways in which a cooperator can be matched withanother cooperator. One way is that the cooperator is correctly labelledas a cooperator and the apparent cooperator that it is matched with is anactual cooperator. The other way is that the cooperator is mislabelled asa defector, but has the good fortune to be matched with a cooperator thathas been mislabelled as a defector. Adding the probabilities of these twooutcomes, we find that

p(x) = αCc(x) + (1− α)Cd(x) (41)

Similarly, a defector can be matched with a cooperator in either of twoways. The defector can be mislabelled as a cooperator and be matched witha correctly labelled cooperator, or the defector can be correctly labelled asa defector and matched with a mislabelled cooperator. Thus we have

q(x) = (1− β)Cc(x) + βCd(x) (42)

We can calculate the index of assortativity, which is

a(x) = p(x)− q(x) = (α + β − 1)(Cc(x)− Cd(x)). (43)

It is straightforward to verify that Cc(0) = Cd(0) = 0 and Cc(1) = Cd(1) = 1and therefore, according to Equation 42, a(0) = 0 and a(1) = 0. For thismodel, in contrast to the examples that we have looked at so far, the index ofassortativity is not constant with respect to x. Applying simple calculus toEquation 42, we find that a′(0) > 0, a′(1) < 0. We also find that a′′(x) < 0for all x between 0 and 1, which implies that a(x) is a concave function, thatslopes upwards at x = 0, reaches a maximum somewhere between 0 and 1,and then slopes downward until it reaches x = 1.

In the special case where α = β, a(x) attains its maximum value of(2α− 1)2 at x = 1/2. Figure 3 shows the qualitative nature of the graph ofa(x) when α = β.

There is a simple intuitive explanation of the fact that a(0) = a(1) = 0.In general, a cooperator is more likely to be matched with a cooperator thanis a defector because a cooperator is more likely to be labelled a cooperatorthan is a defector. But if x is small, so that actual cooperators are rare, theadvantage of being matched with an apparent cooperator is small becausealmost all apparent cooperators are actually defectors who have been mis-labelled. Similarly, when x is close to one, defectors are rare, so that mostapparent defectors are actually cooperators who have been mislabelled. In

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Figure 3: Graph of a(x) where α = β

x0 1

(2α− 1)2

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a(x)

the latter case, even if a defector is labelled a defector, his chance of gettingmatched with a cooperator are good. Thus in the two extreme cases, where xapproaches zero and where x approaches one, the chances of being matchedwith a cooperator are nearly the same for a defector as for a cooperator.

In the case where α > β, so that a cooperator is more likely to be cor-rectly labelled than is a defector, the graph of a(x) remains convex downwardbut the peak occurs at x > 1/2, as in Figure 4.2. Simple calculations showthat in the limiting case as α approaches 1, the peak of the graph occursarbitrarily close to the point where x = 1 and a(x) = β.

Figure 4: Graph of a(x) with α > β

x0 1

a(x)

4.3 Population Dynamics

We can use what we know about the function a(x) to analyze the functionδ(x) that expresses the difference between the average payoff to cooperators.Since a(0) = a(1) = 0, it must be that δ(0) = δ(1) = −c < 0. Thus weknow that cooperators get lower payoffs than defectors in monomorphicpopulations, consisting either entirely of defectors or of cooperators. This

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implies that a population consisting entirely of defectors is locally stableand that a population consisting entirely of cooperators is locally unstable.There remains however, the possibility for a stable polymorphic equilibriumwith some cooperators and some defectors.

Figure 5 graphs δ(x) for a Prisoners’ Dilemma with additive payoffs. Wehave drawn this figure so that δ(x) is positive for some values of x betweenzero and 1. The small arrows show directions of movements strarting fromany intitial position. There are two locally stable equilibria. One stableequilibrium occurs at the point where x = 0. The other is at the pointmarked A. For any level of x to the left of the point B or to the right of thepoint A, δ(x) < 0 and so x, the proportion of cooperators in the population,would decline. For any level of x between the points A and B, δ(x) > 0 andso in this region x would increase.

Figure 5: Graph of δ(x) for additive Prisoner’s Dilemma

← ←→x0 1

δ(x)

−c

AB

For Prisoners’ Dilemma games with additive payoffs, δ(x) = a(x)b − c.We have shown that that a(0) = a(1) = 0, a′(0) > 0, a′(1) < 0, anda′′(x) < 0 for all x between 0 and 1. It follows that δ(0) = δ(1) < 0,δ′(0) > 0, and δ′(1) < 0, and δ′′(x) < 0 for all x between 0 and 1. Thefact that δ′′(x) < 0 on the interval [0, 1] implies that the graph of δ(x) is“single-peaked” as in in Figure 5. Where this is the case, and if δ(x) > 0for some x, there must be exactly one stable polymorphic equilibrium andone stable monomorphic equilibrium with defectors only.

For Prisoners’ dilemma games with non-additive payoffs, it remains truethat δ(0) = δ(1) < 0, δ′(0) > 0, and δ′(1) < 0. It is not necessarily the casehowever that δ′′(x) < 0 on the interval [0, 1], so the graph of δ(x) need notbe single-peaked. In this case, there may be more than one polymorphicequilibrium, but it still must be that if δ(x) > 0 for some x, then there isat least one locally stable polymorphic equilibrium and one locally stable

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monomorphic equilibrium with x = 0.

5 Related Literature

W.D. Hamilton’s classic paper [5] on the evolution of social behavior amongrelatives showed that for some kinds of interactions between relatives, nat-ural selection will lead to a degree of cooperativeness that can be quantifiedin terms of genetic relatedness. In later work, Hamilton [6] [7] observed thatthe theory of kin selection can be viewed as a special case of assortativematching between individuals who are not necessarily related genetically.

Wilson and Dugatkin [12] consider assortative matching in N -playergroups where players are programmed with possible strategies that can bedescribed by a real number x from some specied interval. An individualwith strategy x bears a personal cost cx and confers benefits of bx on everymember of the group, where it is assumed that Nb > c > b.6 The authorspoint out that everyone will prefer to belong to a group consisting of oth-ers with higher x. Therefore if group membership requires mutual consentand if individual x’s are common knowledge, groups would segregate exactlyaccording to their values of x and those with the highest value of x wouldreceive the highest payoffs. They show through simulations that where in-dividual types are imperfectly known so that assortativity is partial, therecan be selection pressure for high individuals with high x.

Myerson, Pollock, and Swinkels [9] introduce a viscosity parameter thatis equivalent to a constant index of assortativity as defined in this paper. Fora population with a finite number of pure strategies, they define a δ-viscouspopulation equilibrium to be an assignment of proportions of the populationplaying each strategy such that for each strategy played by a positive pro-portion, this strategy is a best response to the current strategy mix underthe assumption that an individual will meet its own type with probabilityδ and will meet a randomly selected member of the overall population withprobability 1 − δ. Thus a δ-viscous equilibrium is a symmetric Nash equi-librium for a game in which the payoff to a player is his expected payoff ifwith probability δ he encounters a someone playing the same strategy as hisown and with probability 1− δ he encounters a randomly chosen member ofthe entire population.7

6This game is equivalent to the “voluntary provision of public goods game” that ismuch studied in experimental economics.[8]

7The authors also propose an interesting Nash equilibrium refinement which selects

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Bergstrom [2] shows that for symmetric games between relatives in a pop-ulation of sexual diploids, a monomorphic equilibrium will be stable againstinvasion by dominant mutants and by recessive mutants if each player actsso as to maximize a “semi-Kantian” expected utility function that assignsa probability weight k to the event that one’s opponent mimics one’s ownbehavior and 1− k to the event that one’s opponent is a random draw fromthe population, where k is defined by the genetic coefficient of relatednessbetween the two relatives.

The published papers that seem to be most closely related to this one areEshel and Cavalli-Sforza8 [4] and by Bergstrom and Stark [1]. Both papersstudy models with assortative matching of players in Prisoners’ Dilemmagames and consider examples in which sexual haploid and sexual diploidplayers recognize genetic kin. Eshel and Cavalli-Sforza also study a modelin which players are able to actively choose partners who are likely to treatthem favorably, though their model of partner selection differs from oursin that they emphasize search costs rather than inaccuracy of determiningothers’ types.

The current paper suggests a unifying method for treating a great va-riety of models of assortative encounters and I think offers a simpler, moretransparent method of deriving both known and new results.

References

[1] Theodore Bergstrom and Oded Stark. How altruism can prevail in anevolutionary environment. American Economic Review, 83(2):149–155,1993.

[2] Theodore C. Bergstrom. On the evolution of altruistic ethical rules forsiblings. American Economic Review, 85(1):58–81, 1995.

[3] L.L. Cavalli-Sforza and M.W. Feldman. Cultural transmission and evo-lution: A quantitative approac. Princeton University Press, Princeton,NJ, 1981.

[4] Ilan Eshel and L.L. Cavalli-Sforza. Assortment of encounters and evo-lution of cooperativeness. Proceedings of the National Academy of Sci-ences, 79:1331–1335, February 1982.

only those Nash equilibria that are the limit of δ-viscous equilibria as δ approaches zero.8Toro and Silio [10] offer interesting extensions of Eshel and Cavalli-Sforza’s paper.

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[5] W.D. Hamilton. The genetical evolution of social behavior, parts i andii. Journal of Theoretical Biology, 7:1–52, 1964.

[6] W.D. Hamilton. Selection of selfish and altruistic behavior in someextreme models. In J.F. Eisenberg and W.S. Dillon, editors, Man andBeast: Comparative Social Behavior, pages 57–91. Smithsonian Press,Washington, D.C., 1971.

[7] W.D. Hamilton. Inate social aptitudes of man: an approach from evo-lutionary genetics. In R. Fox, editor, Biosocial Anthropology, pages133–155. Malaby Press, London, 1975.

[8] John Ledyard. Public goods: a survey of experimental research. InJohn H. Kagel and Alvin E. Roth, editors, The Handbook of Experimen-tal Economics, chapter 2, pages 111–181. Princeton University Press,Princeton, N.J., 1995.

[9] Roger B. Myerson, Gregory B. Pollock, and Jeroen M. Swinkels. Vis-cous population equilibrium. Games and Economic Behavior, 3:101–109, 1991.

[10] M. Toro and L. Silio. Assortment of encounters in the two-strategygame. Journal of Theoretical Biology, 123:193–204, 1986.

[11] Jorgen Weibull. Evolutionary Game Theory. MIT Press, Cambridge,MA, 1995.

[12] David Sloan Wilson and Lee A. Dugatkin. Group selection and assorta-tive interactions. The American Naturalist, 149(2):336–351, February1997.

[13] Sewall Wright. Systems of mating. Genetics, 6(2):111–178, March 1921.

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